Question

If $3x^2+10xy+3y^2-15x-21y+k=0$ represents a pair of straight lines, then the value of $k$ is :

• A. $85$
• B. $8$
• C. $-18$
• D. $18$

Answer 1

The correct option is A $85$

$3x^2+10xy+3y^2-15x-21y+k=0\to \left(i\right)$

general equation of straight line is

$a{x}^2+2fgh-b{g}^2-a{f}^2-c{h}^2=0\to$

$abc+2fgh-b{g}^2-a{f}^2-c{h}^2=0\to$

comparing equation 1 with 2, we get

$a=3,h=5,b=3,g=\frac{-15}{2},f=\frac{-21}{2},c=k$

putting all values in equation 3, we get

$\begin{array}{c}3\times 3\times k+2\times 5\times \left(\frac{-15}{2}\right)\times 5-3{\left(\frac{-21}{2}\right)}^2-k{\left(5\right)}^2=0\\ 9k-375-\frac{675}{2}-\frac{1323}{2}-25k=0\\ -16k=\frac{1323}{2}+\frac{675}{2}+\frac{375}{1}\\ -16k=\frac{2748}{2}\\ k=\frac{+2748}{32}\\ k=85.87\end{array}$